C7-C8 Solving three simultaneous equations with matrices Quiz: AQA Further Maths, Unit 2

The 20 questions

1. If A is non-singular and AX = B, what is X?

   • A + B
   • A B^-1
   • A^-1 B
   • B A^-1

2. Geometrically, where do three planes with a unique solution intersect?

   • A parabolic curve
   • A single line
   • A single point
   • Three distinct points

3. How many solutions exist for a linear system whose 3x3 matrix has zero determinant?

   • Zero or infinite
   • Always infinite
   • Always zero
   • Exactly one

4. Solve x + y + z = 6, 2x - y + z = 3, x + 2y - z = 2.

   • x = 1, y = 3, z = 2
   • x = 1, y = 2, z = 3
   • x = 2, y = 1, z = 3
   • x = 3, y = 2, z = 1

5. What is the determinant of [[1, 1, 1], [2, -1, 1], [1, 2, -1]]?

   • 5
   • 1
   • 7
   • -7

6. How are the planes x + y + z = 1, x + y + z = 2, and x + 2y + 3z = 0 arranged?

   • Three parallel planes
   • Sheaf of planes
   • Two parallel planes
   • Triangular prism

7. How do the planes x + 2y + 3z = 6, 2x + 4y + 6z = 12, and x + y + z = 3 intersect?

   • At a point
   • In a line
   • In a plane
   • No intersection

8. For which value of k is the matrix [[1,1,1],[1,2,3],[1,3,k]] non-singular?

   • k ≠ 5
   • k ≠ 0
   • k = 0
   • k = 5

9. For which k is [[1, 2, 3], [2, k, 6], [1, 1, 1]] singular?

   • k = -4
   • k = 4
   • k = 2
   • k = 6

10. Solve x + y + z = 3, x - y = 0, z = 1.

    • x = 1, y = 0, z = 2
    • x = 1, y = 1, z = 1
    • x = 2, y = 2, z = -1
    • x = 0, y = 2, z = 1

11. Solve x + 2y + 3z = 14, y + z = 4, z = 2 by back substitution.

    • x = 6, y = 2, z = 2
    • x = 2, y = 4, z = 2
    • x = 4, y = 2, z = 2
    • x = 4, y = 4, z = 2

12. For which k is [[1, 1, 1], [1, k, 1], [1, 1, 1]] singular?

    • All real k
    • k = 0 only
    • No real k
    • k = 1 only

13. What is the determinant of the coefficient matrix for x - y + z = 1, 2x + y = 3, x + z = 2?

    • -2
    • 2
    • 1
    • 4

14. If matrix A is non-singular, what is the unique solution to AX = B?

    • X = AB⁻¹
    • X = A⁻¹B
    • X = BA
    • X = BA⁻¹

15. Solve x + y + z = 6, x - y + z = 2, x + y - z = 0 for z.

    • z = 2
    • z = 6
    • z = 3
    • z = 1

16. For A = diag(1, 1, 2) and B = (1, 2, 3), solve AX = B.

    • x = 1/2, y = 1, z = 3/2
    • x = 1, y = 2, z = 3
    • x = 1, y = 2, z = 6
    • x = 1, y = 2, z = 3/2

17. What is det [[1, 0, 0], [0, 1, 0], [0, 0, 2]]?

    • 1
    • 0
    • 3
    • 2

18. What is the inverse of the diagonal matrix diag(2, 1, 1)?

    • diag(-2, -1, -1)
    • diag(1/2, 1, 1)
    • diag(1/2, 0, 0)
    • diag(2, 1, 1)

19. What condition ensures that a matrix equation AX = B has a unique solution?

    • det(A) = 1
    • det(A) ≠ 0
    • det(A) = 0
    • det(A) > 1

20. Solve 2x = 4, y - z = 1 and y + z = 5.

    • x = 4, y = 3, z = 2
    • x = 2, y = 3, z = 2
    • x = 2, y = 2, z = 3
    • x = 2, y = 3, z = 4

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